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Re: [ontolog-forum] Ontology, Information Models and the 'Real World':

To: Waclaw Kusnierczyk <Waclaw.Marcin.Kusnierczyk@xxxxxxxxxxx>
Cc: Ontolog Forum <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Sat, 2 Jun 2007 18:21:44 -0700
Message-id: <p06230929c2878e967ca1@[]>
>Pat Hayes wrote:
>>>And, I guess,
>>>"the 'continuous present' indicated by the English 'ing' ending, as in
>>>It is raining, means that the proposition 'rains' is true throughout
>>>some interval containing 'now'" [IKL guide]
>>>should be read as
>>>"the 'continuous present' indicated by the English 'ing' ending, as in
>>>It is raining, means that the false proposition 'rains' stands in the
>>>ist relation to some interval containing 'now'"?
>>Why do you assume that 'rains' is logically false? I have no idea 
>>what its actual logical truth-value is;
>because I take
>(that (rains))
>to be a proposition that it rains, without any indexicals, that is, 
>that it rains everywhere, at all times.    (01)

I don't think that is how it should be understood. After all, that 
can be stated explicitly:    (02)

(forall (c)(ist c (that (rains))))    (03)

>  Or how should it be understood?    (04)

I have no idea. You wrote it, not me :-). In IKL, a relation with no 
arguments is itself a proposition, so one could write it without the 
"that":    (05)

(forall (c)(ist c rains))    (06)

>>but in any case that would be irrelevant to its *contextual* truth, 
>>which is modelled in IKL by the ist relation. "True throughout an 
>>interval" and "standing in the ist relation to an interval" are 
>>just two ways to say the same thing. Or perhaps, if you feel that 
>>truth at a time is something fundamental, by all means say that the 
>>ist-formulation is IKL's way of modelling or describing or 
>>representing the notion of truth at a time.
>But are we not taking a round here, saying that (ist c p) *does* 
>mean that p *is true* in c, even if it is false?    (07)

My point is that truth in a noncontextual classical logic such as FOL 
or CL or IKL - logical truth - on the one hand; and 'truth at a time' 
or 'truth in a context', on the other, are two *distinct* notions. 
They have different meanings and different logics, and support 
different notions of interpretation and of proof. They can be related 
in various ways, which are well-understood, but they are not the same 
notion. In particular, if I can warp the English for a moment and 
write L-truth for the first notion, contextual truth is *not* L-truth 
at a time or L-truth in a context, since it is quite literally 
meaningless to relativize L-truth to a context. So if we look at a 
simple English present tense sentence such as "gusty winds exist", it 
can be understood in two different ways: as uttered in the present 
(and perhaps a particular place, ie with an implicit 'now, here') and 
referring to it, or as uttered timelessly. And these are not the same 
kind of meaning. When setting out to formalize meanings, one has to 
make a choice whether to attempt to model the eternal interpretation 
or the tensed (more generally, contextual) interpretation. One gets a 
different logic in each case.    (08)

So, now, armed with this distinction: IKL, being a classical logic, 
takes the first route. Truth in IKL is understood non-contextually. 
IKL sentences are not asserted "in" a context of any kind. They are 
simply asserted. The English that they model, if that can even be 
spoken of coherently, is only that of 'eternal' sentences in the 
simple grammatical present tense, but understood as tenseless in 
meaning. There is no implicit "here, now" indexical lying behind the 
meaning of any IKL sentence. So, to model the meaning of English 
sentence forms which *are* indexical or contextual in this way in 
IKL, it is necessary to put in these contextual parameters 
explicitly, to "de-contextualize" the logical form. To represent the 
meaning of the New Mexico road sign (omitting the modality for now to 
keep it simple) "Gusty winds exist" in IKL, one would write not    (09)

(exist (x)(and (Gusty x)(Wind x)))    (010)

but something more like    (011)

(exist (x)(and (Location x herePlace)(Time x nowTime)(Gusty x)(Wind x)))    (012)

(There are also several other ways, and this is something of a 
caricature, but you get the idea.)    (013)

So far IKL is indistinguishable from any classical FOL: one has to 
'de-contextualize' any context-dependent assertion or indexical 
content to represent its 'context-relative' meaning in any 
noncontextual logical framework. But IKL also has the ability to 
refer to propositions, so this gives us yet another way to encode the 
intended meaning of a contextual assertion, one which is 
superficially very similar to the way it is represented in a 
contextual logic. We can in fact mirror the context-logical syntax 
almost exactly. Context logic has a basic expression form    (014)

a:   (ist C <sentence>)    (015)

eg. (ist Present (exists (x) (and (gusty x)(wind x))) )    (016)

where 'ist' is a logical symbol on a par with the connectives and 
quantifiers, one with a meaning fixed by the context-logical 
semantics. As it stands this is not well-formed in IKL, since it 
seems to state a relation between a thing and a sentence. But we can 
mirror this in IKL with the closely similar expression    (017)

[a]:  (ist C (that <sentence>) )    (018)

eg. (ist C (that (exists (x) (and (gusty x)(wind x)))) )    (019)

which is syntactically legal: it states that a relation called "ist" 
(which is now simply a relation symbol, not a special logical form) 
holds between C and the proposition (that <sentence>). What it 
*means* however is essentially aribitrary in IKL. In particular, it 
has no IKL-sanctioned *logical* connection with truth in IKL. But we 
can agree to give it a meaning, and it is useful to do so.    (020)

Consider the translation from a to [a] to be an embedding of context 
logic into IKL syntax. Now, it turns out that this translation 
completely mirrors the context logic. That is, A entails B in context 
logic just when [A] logically entails [B] in IKL. (Actually this is 
not quite true: the syntactic embedding needs to also perform a 
mapping on the names used opaquely inside contexts, so is more 
complex than described here: see the IKL guide for details.)    (021)

In context logic, "ist" defines a context-relative notion of truth. 
In IKL, "ist" is simply a relation which holds between C and the 
proposition (that <sentence>) precisely when <sentence> is true-in-C 
in the context logic. I tend, perhaps confusingly, to use this 
terminology when referring to the IKL translation.    (022)

(There is some history here which may be relevant. IKL was developed 
in order to provide an interlingua for a variety of formalisms, two 
of which used contexts and three did not. It was therefore necessary 
for us to go into some detail concerning the various notions of truth 
and satisfiability involved. John McCarthy, who introduced context 
logic some time ago but did not work out the underlying theory in 
detail, insisted that the sentence form inside 'ist' should be 
regarded as denoting a proposition, and 'ist' thought of as a 
relation between contexts and propositions, and that contexts were 
simply whatever satisfied whatever context theory one had in place. 
All of which lay behind our design of IKL. Having introduced 
propositions as objects, however, we discovered, rather to our own 
surprise, that this rendered the rest of the context logic 
irrelevant: one can express the entire context-logical framework 
within a classical logic which can describe propositions. (It also 
seems, although we have not yet fully investigated this, to provides 
a new way of resolving the classical paradoxes.) As classical logics 
are far better understood than contextual logics, this seems like a 
useful technical advance.)    (023)

I believe you may feel that truth is often, or perhaps always, best 
understood as being relative to a context, so that truth-in-a-context 
is an objective matter and context logic is the most appropriate 
formal vehicle for meaning. Although this position is widely held, I 
myself do not agree: I understand the tensed use of the present tense 
as referring to the actual present time, so that "It is raining now" 
spoken at 3.30 pm on the 21 March 1998 is exactly and precisely 
synonymous with the tenseless assertion "Raining at 3.30 pm on the 21 
March 1998". Contextually asserted and indexical assertions are, on 
this view, simply incomplete, and have part of their logical meaning 
implicit. Most of what is said in natural language does not express a 
determinate proposition until it has been rendered non-contextual by 
having all its implicit indexicality resolved, and at that point it 
can perhaps be rendered into a classical logic, but not before.    (024)

Hope this helps.    (025)

Pat    (026)

>vQ    (027)

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